Az Eszterházy Károly Tanárképző Főiskola Tudományos Közleményei. 1998. Sectio Mathematicae. (Acta Academiae Paedagogicae Agriensis : Nova series ; Tom. 25)

MÁTYÁS , F., Bounds for the zeros of Fibonacci-like polynomials

20 Ferenc Mátyás two distinct Gershgorin-circles. The midpoints of these circles are — 6 and 0 in the Gaussian plane, while by (7) their radii are jaj and 2, respectively, i.e. they are the sets (circles) Ii\ and K

2 . (We omitted the circle with midpoint 0 and radius 1, because this circle is contained by one of the above circles.) Since G

n(fl, x + b, x) is the characteristic polynomial of the matrix A

n, and A

N

L

, X

N

2,..., A

N

N are the zeros of it so from (8) and (9) we get that 5 ? • • • J ^nn G U Ii 2 . This completes the proof. Proof of the Main Result. We have seen in the proof of Theorem 2 that the Gershgorin-circles K\ and K

2 don't depend on n if n > 2, therefore for any n > 2 the zeros of the polynomials G

n(a, x + b, x) belong to the sets (circles) Ii 1 and Ii

2. I.e. if G

n(a,x + b.x) = 0 for a complex x, then (15) \x\ < max(|a| + |6| ,2). Since Gi(a,x + b,x) = 0 if x = -b therefore (15) also holds if n = 1. This completes our proof for every integer n > 1. References [1] V. E. HOGGAT, JR. AND M. BICKNELL, Roots of Fibonacci Polynomials, The Fibonacci Quarterly 11.3 (1973), 271-274. [2] HONGQUAN YU, YI WANG AND MINGFENG HE, On the Limit of Generalized Golden Numbers, The Fibonacci Quarterly 34.4 (1996), 320-322. [3] F. MÁTYÁS, Real Roots of Fibonacci-like Polynomials, Proceedings of Number Theory Conference, Eger (1996) (to appear) [4] F. MÁTYÁS

, The Asymptotic Behavior of Real Roots of Fibonacci-like Polynomials, Acta Acad. Paed. Agriensis, Sec. Mat., 24 (1997), 55-61. [5] G. A. MOORE, The Limit of the Golden Numbers is 3/2, The Fibonacci Quarterly 32.3 (1994), 211-217. [6] H. PRODINGER, The Asymptotic Behavior of the Golden Numbers, The Fibonacci Quarterly 35.3 (1996), 224-225. [7] P. E. RICCI

, Generalized Lucas Polynomials and Fibonacci Polynomials, Riv. Mat. Univ. Parma (5) 4 (1995), 137-146. FERENC MÁTYÁS KÁROLY ESZTERHÁZY TEACHERS' TRAINING COLLEGE DEPARTMENT OF MATHEMATICS H-3301 EGER, PF. 43 HUNGARY E-mail: matyasiekti.hu